Suppose $X_1,\dots,X_n$ are independent Gaussians, where $X_k \sim N(\mu_k,1)$. I am interested in $$ Z \stackrel{\rm def}{=} \max_{1\leq k\leq n} X_k $$ and specifically on the asymptotics of $\mathbb{E}[Z]$ (as a function of $n$ and $(\mu_k)_k$), and the concentration around this expected value.

The case where all $\mu_i$'s are equal is of course well-understood (equivalent to all $X_i$'s being $N(0,1)$); but the proofs I know do not seem to generalize to yield anything usable.

As a maybe simpler case, what about having $\mu_1=\dots=\mu_{n-1}=0$ and $\mu_n \neq 0$? *(where $\mu_n$ may or may not depend on $n$, depending on what one can prove; I am thinking of it as a small constant)*

**Following a comment below:** even in the "simpler" case, what I would like is to understand the gap between the above and the standard "all means are zero" cases (even only for the expected value, setting aside the concentration around it). That will be in the second-order term (or even lower?) of the asymptotics, since the leading term should still be $\sqrt{2\log n}$ for constant $\mu_n$.

simpler case, do you assume $\mu_n=O(1)$? Then it should be easy to see that the effect of $\mu_n$ is negligible. $\endgroup$ – Iosif Pinelis Nov 27 at 19:25